Sum of Normal Random Variable

[corpact90]

Hi everyone, I have a doubt regarding normal random variables.
I know it could really easy to solve but still I can't figure this out.
So, if you have to sum to gaussian variable as follows:
aN(0,1)+bN(0,1)
where a and b are constant we can proceed as follows:
aN(0,1)+bN(0,1)=N(0,a^2)+N(0,b^2)=N(0,a^2+b^2)

why is this different from this:
aN(0,1)+bN(0,1)=(a+b)N(0,1)=N(0,(a+b)^2)

which one of the two approaches is correct?

Thanks

 

[djjain80]

1st approach is correct..coz N(0'1) is a distribution variable but in second approach its been assumed as algebric variable...
so its not sum of distribution but its sum of variable of a distribution that has algebric properties

 

[John Lee]

Since \(a^2 + b^2 \neq (a + b)^2 \)